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Bulletin of the London Mathematical Society 1987 19(3):259-263; doi:10.1112/blms/19.3.259
© 1987 by London Mathematical Society
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© Oxford University Press

A Completeness Criterion for Inner Product Spaces

J. Hamhalter and P. Pták

Technical University of Prague – El. Eng., Department of Mathematics 166 27–Prague 6, Czecheslovakia

We show that a separable inner product space is complete if and only if its lattice of strongly closed subspaces possesses at least one state. This gives a measure-theoretic characterization of Hilbert spaces among inner product spaces and, as a by-product, exhibits a ‘continuous’ example of a stateless orthocomplemented lattice.


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